LCM Calculator: Find The Least Common Multiple With Steps

The least common multiple comes up whenever you add fractions, line up repeating schedules or split items into equal packs. This guide covers four methods, the link between the LCM and the GCF, and the edge cases that trip students up. Every example was checked with Python's math library.

Note: Calc Solver (calc-solver.co) is an independent guide site and is not affiliated with any calculator mentioned here.

What The Least Common Multiple Means

A multiple of a number is that number times 1, 2, 3 and so on. A common multiple is a number that shows up in the multiple lists of two or more numbers. The least common multiple is the smallest one.

For 4 and 6:

• Multiples of 4: 4, 8, 12, 16, 20, 24
• Multiples of 6: 6, 12, 18, 24, 30

The common multiples are 12, 24, 36 and so on. The smallest is 12, so LCM(4, 6) = 12.

The LCM is also called the lowest common multiple. When the numbers are denominators of fractions, the same value is called the least common denominator (LCD).

How To Use The LCM And GCF Calculator

The calculator above finds both values at once, because they are closely linked.

1. Pick a tab. Use LCM or GCF for whole numbers, or LCD of Fractions to add or compare fractions.
2. Type your numbers. Separate them with commas or spaces, like 12, 18, 30. You can enter up to 15 numbers.
3. Choose a method. Pick the one your teacher uses so the steps match your homework.

Four Ways To Find The LCM By Hand

Each method gives the same answer. Listing is easiest for small numbers, prime factorization and the ladder work for any size, and the GCF formula is fastest for two numbers.

Method 1: List The Multiples

Write out multiples of each number until one appears in every list. For 10 and 8:

• Multiples of 10: 10, 20, 30, 40
• Multiples of 8: 8, 16, 24, 32, 40

LCM(10, 8) = 40. A time saver: list multiples of the largest number only, and check each one against the smaller numbers. 10, 20 and 30 fail for 8, but 40 works.

Method 2: Use Prime Factorization

Break each number into primes. Then take every prime that appears, at its highest power, and multiply.

For 24 and 36:

LCM = 2³ × 3² = 8 × 9 = 72.

If you need help breaking numbers into primes, the prime factorization tab in our factor calculator shows a factor tree for each number.

Method 3: The Ladder (Cake) Method

Write the numbers in a row. Divide by any prime that goes into at least two of them, and carry down any number it does not divide. Stop when no prime divides two numbers. Then multiply everything on the left and along the bottom.

For 12, 18 and 30:

No prime divides two of 2, 3 and 5, so stop. LCM = 2 × 3 × 2 × 3 × 5 = 180.

The ladder gives the GCF for free: multiply only the left side, 2 × 3 = 6.

Method 4: Divide By The GCF

For two numbers, multiply them and divide by their greatest common factor:

LCM(a, b) = (a × b) ÷ GCF(a, b)

For 24 and 36: the GCF is 12, so LCM = (24 × 36) ÷ 12 = 864 ÷ 12 = 72.

This is the fastest method for large numbers, because the GCF is quick to find with the Euclidean algorithm. Divide the larger number by the smaller and keep the remainder: 36 ÷ 24 leaves 12, then 24 ÷ 12 leaves 0. The last nonzero remainder, 12, is the GCF.

How To Find The LCM Of Three Or More Numbers

To find the LCM of three or more numbers, use prime factors or the ladder, or work in pairs. The pair method finds the LCM of the first two numbers, then the LCM of that result with the next number.

For 12, 16 and 21:

1. LCM(12, 16) = 48
2. LCM(48, 21) = 336

So LCM(12, 16, 21) = 336.

A classic example is the LCM of every number from 1 to 10. Using the highest prime powers, 2³ × 3² × 5 × 7 = 2,520. That means 2,520 is the smallest number that 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10 all divide into evenly.

One warning: the GCF formula from Method 4 does not work for three numbers. For 2, 4 and 6, the product is 48 and the GCF is 2, which gives 24. The real LCM is 12. Stick to pairs or prime factors.

LCM Vs GCF: What Is The Difference?

The LCM is the smallest number your numbers divide into. The GCF is the largest number that divides into all of them. The LCM is always at least as big as the largest number, and the GCF is never bigger than the smallest.

For two numbers, LCM × GCF always equals the product of the numbers. For 24 and 36: 72 × 12 = 864, and 24 × 36 = 864.

Using The LCM To Add And Subtract Fractions

When fractions have different denominators, the LCM of the denominators is the least common denominator. Using it keeps the numbers small and saves a simplifying step at the end.

Add 5/12 + 7/18:

1. LCM(12, 18) = 36, so the LCD is 36.
2. 5/12 = 15/36, because 36 ÷ 12 = 3 and 5 × 3 = 15.
3. 7/18 = 14/36, because 36 ÷ 18 = 2 and 7 × 2 = 14.
4. 15/36 + 14/36 = 29/36.

Multiplying the denominators (216) also works, but then you must reduce 174/216 to 29/36. For mixed numbers and longer problems, the fraction calculator shows each step.

LCM Word Problems You Will See

LCM word problems ask when two repeating things line up, or the smallest amount that splits evenly into different group sizes. Here are the two most common types.

Packs that do not match. Hot dogs come in packs of 10 and buns come in packs of 8. What is the fewest of each you can buy so none are left over? LCM(10, 8) = 40. Buy 40 of each: 4 packs of hot dogs and 5 packs of buns.

Schedules that repeat. One bus leaves every 12 minutes and another every 15 minutes. Both leave at 8:00 a.m. When do they leave together again? LCM(12, 15) = 60, so at 9:00 a.m.

If a problem asks for the largest equal group instead, you need the GCF.

Special Cases: Coprime, Zero, Negatives And Decimals

A few kinds of numbers follow simple rules, and knowing them saves time.

Decimals: multiply every number by the same power of 10 to make them whole, find the LCM, then divide back. For 0.4 and 0.6, multiply by 10 to get 4 and 6. LCM(4, 6) = 12, and 12 ÷ 10 = 1.2. Check: 1.2 ÷ 0.4 = 3 and 1.2 ÷ 0.6 = 2.

How To Find The LCM On A TI-84 Or Casio Calculator

Some calculators have an LCM function built in, while others do not.

TI-84 Plus CE: press MATH, move right to NUM, and choose 8: lcm(. Type two numbers separated by a comma, close the parenthesis and press ENTER. The same menu has 9: gcd( for the GCF, as shown in Texas Instruments' MATH menu guide. It takes two numbers at a time, so nest it for three: lcm(lcm(12,16),21).

Casio fx-991CW ClassWiz: choose LCM from the Numeric Calc menu, type the numbers separated by a comma and press equals. Casio's fx-991CW manual walks through the same keys. The older fx-991EX has no LCM function, so use the ladder or the GCF formula there.

Common LCM Mistakes

Most LCM errors come from mixing it up with the GCF or using the wrong prime powers.

Summary

• The LCM is the smallest number that every number in a list divides into evenly.
• List multiples for small numbers. Use prime factors (highest powers) or the ladder for bigger ones.
• For two numbers, LCM = (a × b) ÷ GCF. For three or more, work in pairs.
• The LCM of denominators is the least common denominator for fractions.
• Coprime numbers have an LCM equal to their product.

Next step: enter your numbers in the LCM and GCF calculator at the top of this page. For more number tools, see the math solver or the Calc Solver homepage.

Frequently Asked Questions